arXiv research

A mathematical framework connects neural networks and polynomial regression for better model understanding.

We develop a comprehensive mathematical framework for polynomial jump-diffusions in a semimartingale context, which nest affine jump-diffusions and have broad applications in finance. We show that the polynomial property is preserved under polynomial transformations and Lévy time change. We present a generic method for…

Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.

New framework allows reinforcement learning with polynomial sample complexity.

F. Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the sl(n)-link invariant (a one…

New link polynomials linked to cluster theory.

Survey on using low-degree polynomials to assess statistical tasks complexity.

Proposes a model for long-term electricity contracts with explicit computation and easy calibration.

Polynomial-time DP algorithm for learning Gaussians with matching sample complexity.

The SO(3) Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks are included.

Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.

The paper studies SDP feasibility and sos ranks for specific polynomials.

New framework uses score-based priors to solve ill-conditioned polynomial equations, improving signal recovery from noisy data.

Polynomial processes in Banach spaces via infinitesimal generator and ODEs.

Develops a braid-theoretic framework to analyze chirality in molecular knots.

Unified approach to experimental design using interlacing polynomials.

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inferen…

This paper improves flow models to better handle perturbations in real-world data.

nn2poly converts neural networks into interpretable polynomial models.

Relative Thom polynomials for maps around boundaries established.

Classifies small links in an unmarked solid torus.

Combines Kleinian groups and polynomials into a dynamical system.

This work improves polynomial approximations for functions with asymmetric behavior.

Study introduces a new method for multiple parameter regularization in polynomial functional regression.

Novel Jones polynomial for open curves in 3D space.

In the setting of polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for p…

Polynomial Chaos Expansion improves operator learning for PDEs.

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ of the Thompson groups $F$ and $T$, respectively, together with a procedure that associates an oriented link diagram to any element o…

LiPopt uses polynomial optimization to estimate neural network Lipschitz constants efficiently.

MAP perturbation models have emerged as a powerful framework for inference in structured prediction. Such models provide a way to efficiently sample from the Gibbs distribution and facilitate predictions that are robust to random noise. In this paper, we propose a provably polynomial time randomized algorithm for learn…

Infinite dimensional measure-valued processes modeled as polynomial diffusions.

HiPPO framework optimizes memory compression for sequential data.

Enhanced PC$^2$ improves surrogate modeling for high-dimensional problems.

Study disproves conjecture about low-degree polynomials in hypothesis testing.

Holonomy-preserving transformations help recover Alexander polynomials from graph zeta functions.

A method for interpreting SVMs using polynomial kernels, revealing model complexity.

New theory for unoriented virtual links, extending classical invariants.

New findings on computational limits for estimating hidden structures.

New framework uses unsupervised learning for efficient exploration in RL.

New framework mated Kleinian groups with complex polynomials, revealing unique group properties.

The problem of high-dimensional path-dependent optimal stopping (OS) is important to multiple academic communities and applications. Modern OS tasks often have a large number of decision epochs, and complicated non-Markovian dynamics, making them especially challenging. Standard approaches, often relying on ADP, dualit…

New algorithm reduces dynamic regret for noisy gradient feedback with piecewise polynomial comparators.

Estimates hybrid dynamical systems with polynomial expansions and Markovian switching.

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables tha…

This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial χ(Q) of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chroma…

In this paper we give a quantum statistical interpretation for the bracket polynomial state sum <K> and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to al…

Polynomial-time tester-learner for general halfspaces with Gaussian adversarial noise.

In this paper, we investigate the capability of the universal Kriging (UK) model for single-objective global optimization applied within an efficient global optimization (EGO) framework. We implemented this combined UK-EGO framework and studied four variants of the UK methods, that is, a UK with a first-order polynomia…